It remains to prove that \( \mathcal {S}_\alpha \mathcal {R}_{\theta _0} u \)  is a
sensible solution to the state equation of problem eq:model for each \( u\in L^2(0,T;Z) \) . To this end, we first introduce the very weak solution concept
of the state equation, following the idea in [1]}. For any \( 0 <\alpha <1 \)  and \( g \in L^2(0,T;Y) \) , there exists a unique \( w \in {}^0H^\alpha (0,T;Y) \cap L^2(0,T;X) \)  such that (cf. [2]}, [3]})
\(({T-}^\alpha - \mathcal {A}) w = g\) 
